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Financial Networks as
Probabilistic Graphical
Models (PGM)
CAMBRIDGE, SEPTEMBER 2015
What PGMs are
A set of random variables can be given a graphical representation which
encodes the conditional independencies between them in a visually
appealling form
The name of this representation is Probabilistic Graphical Models (PGM)
A graphical representation consists of
Nodes – the random variables
Edges – the probabilistic interactions between them
A short taxonomy
There are many types of Probabilistic Graphical Models
Some of them are suitable for studying networks of firms such as:
Bayesian Nets (BN)
Markov Random Fields (MRF)
Chain Graphs (CG)
Directed Cyclic Graphs (DCG)
Probabilistic Graphical Models
Markov Random Field (MRF)
(B ⊥ C | A, D)1
(D ⊥ A | B,C)
A
B
Bayesian Net (BN)
(B ⊥ C | A),
(C ⊥ D | A),
(B ⊥ D | A)
C
D
B
D
D
A
A
C
Chain Component
1. The symbol ⊥ denotes independence relationship
The symbol | denotes “given”. We more compactly want
to say B is independent of C given A and D
Chain Graph (CG)
The set of variables which remain connected
by undirected edges after removing the
directed edges
NB Also D and A are chain components formed
each by 1 node
B
C
(B , C ⊥ D | A)
PGM and Financial Networks
The nodes in a PGM can represent random variables characterising a set
of financial firms and the edges how these variables influence each other
Typical examples of random variables are:
Probabilities of default
Asset returns
Equity returns
Etc
MRF and Default Configurations
A network of firms resembles (with an
oversimplification) a grid of atoms whose debt
interactions are, on the face of it, similar to the those of
the Ising model usually modelled through a MRF
In such a description, firms themselves (and the
interactions between them) can be seen as atoms,
where the default of a set of debtors to firm i can “flip”
i into default
MRFs provide a natural representation of a network of
debt relations, as they are endowed with desirable
screening properties. In fact, if two firms are not
indebted with each other, we have no reason to
believe that they should exert any direct influence on
each other’s probability of default
PGM and Networks of Defaults
Markov Random Field (MRF) representation of a network of probabilities of default
PD3
PD2
PD1
Debt Relationhsips
PD4
PD7
PD5
PD6
Populating the MRF
When specifying a MRF we must define the affinity between the two
variables and we shall do this through a potential ϕ(A, B)
A potential is a positive real-valued function over a discrete domain Ω
ϕ(Ω): Ω → R+
For the boolean MRF below we have to define four values for the potential
ϕ(A, B), ϕ(nA, B), ϕ(A, nB) and ϕ(nA, nB)
Example
A
B
𝐴
𝐴
𝐴
𝐴
B
𝐵
𝐵
𝐵
50
10
20
1
Normalizing
The joint probability is given by:
𝑃 𝐴, 𝐵 =
1
𝜑(𝐴, 𝐵)
𝑍
With the normalization constant:
𝑍=
𝜑(𝐴, 𝐵)
𝐴,𝐵
Example: Joint Probability Table
𝐴
𝐴
𝐴
𝐴
𝐵
𝐵
𝐵
𝐵
61.7%
12.3%
24.7%
1.2%
The Joint Probability Table
When extending to more complex networks what we need to provide is
only the potential encoding the interaction of a random variable with its
neigbors
From local assignments of potentials we build a global characterization of
the network given by the joint probability table
Example: JPT for 3 nodes
Calibration
Such network can be calibrated by knowing 2 sets of quantities
For each 𝑋𝑖 the marginal probability 𝑃(𝑋𝑖 )
For each pair 𝑖, 𝑗 the correlation ρ(𝑋𝑖 , 𝑋𝑗 )
Or:
For each 𝑋𝑖 the marginal probability 𝑃(𝑋𝑖 )
For each pair 𝑖, 𝑗 the joint probability 𝑃(𝑋𝑖 , 𝑋𝑗 )
Or:
For each 𝑋𝑖 the marginal probability 𝑃(𝑋𝑖 )
For each pair 𝑖, 𝑗 the conditional probability 𝑃(𝑋𝑖 |𝑋𝑗 ) (or 𝑃(𝑋𝑗 |𝑋𝑖 ))
Distribution of Defaults
Once the network is calibrated then a network default distribution can be
calculated
Distribution of Losses
…and a distribution of losses in the system
Distribution of Losses
Systemic risk indicators can be computed from the distribution of losses
e.g.:
VaR
Conditional VaR
The contribution of a single firm to the the distribution of losses can be
measured as:
𝑁
𝑖=1 𝑉𝑎𝑅𝑖 )
Component VaR (𝑉𝑎𝑅 =
Component Conditional VaR (𝐶𝑉𝑎𝑅 =
𝑁
𝑖=1 𝐶𝑉𝑎𝑅𝑖 )
Other Networks
The previous setup assumes that the structure of default relationships
between firms is known
Sometimes obtaining such information may be very difficult even for
central banks
Other types of network relationships can be still studied e.g. equity returns
Training a Continuous MRF
If a variable in a network represents
the equity return of a firm one can
train a MRF on a dataset containing
the equity returns of a set of firms1
The learning algorithm will
automatically find the conditional
independencies and detect the
significant edges
One can then transform the equity
returns in network default
probabilities by introducing a default
threshold and discretising
1 An example from Ahelegbey and Giudici (2014)
Reducing the network
One can condition the equity
returns on a global market risk
factor (e.g. the S&P 500 or a
liqudity index) and train a PGM
on the returns residuals which
are non-explained by the risk
factors
The number of links is greatly
reduced but not eliminated
A chain graph approach
The example of the two
previous slides are essentially a
Chain Graph PGM
A Chain Graph of 3 factors F={F1,F2,F3} and 3 firms C={C1,C2,C3}
A Bayesian Net approach
As an alternative one can
choose to train a Bayesian Net
on the firms’residuals as done in
Kitwiwattanachai (2014) on CDS
data of large banks
Δ𝑙𝑜𝑔𝑆𝑖,𝑡 = 𝛼𝑖 + 𝛽𝑚 𝑅𝑚,𝑡 + 𝛽𝑣 Δ𝑉𝐼𝑋𝑡 + 𝜀𝑖,𝑡
where Δ denotes weekly changes of
the CDS 𝑆𝑖,𝑡 of institution 𝑖, 𝑅𝑚,𝑡 is the
S&P500 weekly returns at time t,
and VIX is the CBOE implied volatility index
Conclusions
PGMs are a good framework to modelling financial networks as they can
take into account the inherent stochasticity of complex systems of
interacting entities
PGMs can express conditional independencies as the ones observed in
real network of interacting entities
PGMs can be trained automatically on data to unveil the underlying
structure of the network at hand
Books
Portfolio Management under Stress
A Bayesian Net Approach to Coherent Asset Allocation
Riccardo Rebonato, Alexander Denev
Probabilistic Graphical Models in Finance
Alexander Denev
